Understanding mathematical explanation for Chudakov Effect

I'm trying to better understand some properties of QCD and currently I'm looking into understanding color coherence. I've got the book "Elementary Particle Physics: Foundations of the Standard Model, Volume 2" which is also available on google books.

In the chapter on color coherence (Section 10.2), the QCD is related to the QED Chudakov Effect. I'm trying to understand the mathematical explanation presented in that section, however, I fail to understand what is done, especially since the derivation is full of (hidden) approximations. There's two steps which I can't figure out. Furthermore, I've got some questions regarding the implication. I've linked the full book above and I'll walk you through the derivation as a I understand it. I've highlighted my specific questions as quotes. Hopefully a seasoned QCD calculator can shed some light :-).

The effect happens in the process of $\gamma \to e^+e^- \to e^+e^-\gamma$ which followed the diagram below:

The electron/positron are labelled as $i$ / $j$. Derivation starts by saying that the electron produced at the splitting vertex of the initial photon is virtual and de-excites by emitting the final-state photon with momentum $k$. Afterwards, the real electron has momentum $p$. Two approximations are listed:

1. $|k| \ll |p|$, and
2. $\theta_i = \cos^{-1}(k \cdot p / |k \cdot p |) \to 0$ (small angle approximation?)

1. Energy Imbalance

Now, the energy difference (energy imbalance?) between the virtual and real final state is

$\Delta E = \sqrt{p^2 + m^2} + |k| - \sqrt{(p+k)^2 + m^2} \sim |k|\theta_i^2$

I understand the first part of the equation, however I fail to see how to obtain the approximation. I either end up with many more terms or a $\Delta E = 0$ depending on how strongly I interpret the two approximations listed beforehand.

What are the steps to obtain the approximated energy difference?

Using the Heisenberg uncertainty principle, this energy difference is related to the lifetime of the virtual electron:

$\Delta t \sim \frac1{\Delta E} \approx \frac1{|k|\theta_i^2} = \frac1{|k_T|\theta_i} \approx \frac{\lambda_T}{\theta_i}$

where $\lambda_T$ is the wave length of the photon transverse to the electron (?).

2. Separation of $e^+e^-$

According to the book, the $e^+e^-$ separate by $\Delta d$ during the timespan $\Delta t$:

$\Delta d = \Delta t \cdot \theta_{ij} = \lambda_T \frac{\theta_{ij}}{\theta_i}$

how does this relation work? Presumably the small angle approximation is used somehow, but I fail to understand how the $e^+e^-$ separate by one unit of $\theta_{ij}$ per unit of time.

How is this formula for the separation $\Delta d$ derived?

The argument for the Chudakov effect is then that $\Delta d$ must be larger than $\lambda_T$ since the emitted photon can not distinguish the two otherwise which would imply that it would radiate off a electrically neutral object. Therefore, $\theta_i < \theta_{ij}$.

• How does this change when $\theta_{ij}$ is not small (i.e. the $e^+e^-$ pair has appreciable opening angle)? The small angle approximation is used but is this just to enable a quick calculation? I guess as the opening angle gets larger, the distance $\Delta d$ grows much more quickly, so this might enable more large-angle photon radiation (which is consistent with the result predicted above). However, can this actually be inferred from the calculation presented here?
• As I understand it, this scales with the energy of the photon, i.e. a slightly harder photon may be emitted at a slightly larger angle $\theta_i$. In other words, for two photons $a$ and $b$ with $|k|^{(a)} > |k|^{(b)}$ we can expect that the maximum realistic $\theta_i$ is also following $\theta_i^{(a)} > \theta_i^{(b)}$. Is this correct?
• Does this calculation say anything about whether emitting the photon between the $e^+e^-$ is more likely than outside of their cone?